$GL(n)$-dependence of matrices

TL;DR

This work introduces $\operatorname{GL}(n)$-dependence for matrices in $\mathcal{M}_{n\times m}$, proving that any $m+1$ such matrices are $\operatorname{GL}(n)$-dependent, which generalizes both the classical fact that $m+1$ vectors in an $m$-dimensional space are linearly dependent and the transitivity of the $\operatorname{GL}(n)$-action. The finite-field proof uses a subspace $H\subseteq \mathcal{M}_{n\times n}$ with $\dim H=n$ and $H\subseteq \operatorname{GL}(n)\cup\{0\}$ to obtain a nontrivial relation via a dimension-counting map $f: H^{m+1}\to \mathcal{M}_{n\times m}$. The infinite-field case adopts a double induction on $(n,m)$, with an iterative correction of coefficient matrices by a scalar parameter $x$—choosing $x$ to avoid finitely many forbidden values—so that all coefficients eventually lie in $\operatorname{GL}(n)$. The results are reformulated in terms of subspace row-spaces, showing that any $m+1$ subspaces of dimension at most $n$ in $K^m$ are $\operatorname{GL}(n)$-dependent, linking matrix- and subspace-structures and offering a unified view with potential implications for complexity theory (e.g., KRW conjecture).

Abstract

We introduce the notion of $GL(n)$-dependence of matrices, which is a generalization of linear dependence taking into account the matrix structure. Then we prove a theorem, which generalizes, on the one hand, the fact that $n+1$ vectors in an $n$-dimensional vector space are linearly dependent and, on the other hand, the fact that the natural action of the group $GL(n,{\cal K})$ on ${\cal K}^n\setminus\{0\}$ is transitive.

Theorems & Definitions (14)

• Definition 1
• Theorem 1
• Lemma 1: GJJR10
• Proposition 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 2